Finite Group Databasebeta

← #20276409

Subgroups — #20276409

Order 6784.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811194C211yesmaximal, characteristic
2#16810607C211yesmaximal, characteristic
3#16810421C214yesmaximal, complemented
4C53×#413C211yesmaximal, characteristic
5C22×Dic106C411yescharacteristic
6C2×#11719917—21no—
7#13133215C2212yes—
8#13133204C412yes—
9#13133198—21no—
10#13133198—21no—
11#13133196—21no—
12#13133196—21no—
13#13133176C411yescharacteristic
14#13133136C2212yes—
15#13133134—21no—
16C2×#41×C53C2211yescharacteristic
17C2×M16×C53—21no—
18#101×C53C2212yes—
19#100×C53—21no—
20#41×C53—41no—
21#41×C53—41no—
22C2×Q8×C53C2311yescharacteristic
23C2×D4×C53C2×C411yescharacteristic
24C2×D4×C53D411yescharacteristic
25C2×D4×C53D411yescharacteristic
26C22×C4×C53C2×C411yescharacteristic
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×Dic53—21no—
30C2×Dic106—21no—
31C2×Dic106—41no—
32C2×Dic106—41no—
33C2×Dic106C2×C412yes—
34M16×C53—22no—
35M16×C53—41no—
36C2×C8×C53—41no—
37C22⋊Dic53—21no—
38C22⋊Dic53—22no—
39C22⋊Dic53—22no—
40C4⋊Dic53—21no—
41#11719919—22no—
42#11719919—22no—
43C4×Dic53C2×C412yes—
44#11719917—22no—
45#11719917—41no—
46C2×#88322—41no—
47C23×C53C22⋊C411yescharacteristic
48C23×C53C22⋊C411yescharacteristic
49C23×C53C22⋊C411yescharacteristic
50Q8×C53—41no—
51D4×C53—41no—
52D4×C53—41no—
53D4×C53—41no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53C22⋊C411yescharacteristic
61C2×C4×C53C22×C411yescharacteristic
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×Dic53—22no—
65C2×Dic53—22no—
66C2×Dic53—42no—
67Dic106—41no—
68Dic106—41no—
69Dic106—42no—
70C8×C53—42no—
71#88322—42no—
72C22×C53—21no—
73C22×C53—21no—
74C22×C53—21no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53C2×C22⋊C411yescharacteristic
79C4×C53—21no—
80C4×C53—21no—
81C4×C53—21no—
82C4×C53#9911yescharacteristic
83C4×C53#9911yescharacteristic
84Dic53—42no—
85Dic53—42no—
86#2135—531nomaximal, Sylow
87C2×C53—21no—
88C2×C53—41no—
89C2×C53—41no—
90C2×C53—41no—
91C2×C53C2×#9911yescharacteristic
92#536—531no—
93#413—531no—
94#413D5311yescharacteristic
95#356—534no—
96C53#213511yescharacteristic, complemented, Sylow
97C2×#41C2×D5311yescharacteristic
98C22×Q8—531no—
99C2×M16—21no—
100C2×M16—1061no—
101C4⋊Q8—532no—
102#124—532no—
103#123—1061no—
104#123—1061no—
105C22⋊Q8—1061no—
106C22⋊Q8—1061no—
107#117—531no—
108#101—532no—
109#101C2×D5312yes—
110#100—21no—
111#100—1061no—
112#41—41no—
113#41—41no—
114C2×Q8—532no—
115C2×Q8—1061no—
116C2×Q8—2121no—
117C2×Q8—2121no—
118C2×Q8C22×D5311yescharacteristic
119C2×D4C4×D5311yescharacteristic
120C2×D4D21211yescharacteristic
121C2×D4C22⋊D5311yescharacteristic
122C22×C4—1061no—
123C22×C4C4×D5311yescharacteristic
124C22×C4D21211yescharacteristic
125C22×C4C22⋊D5311yescharacteristic
126M16—22no—
127M16—41no—
128M16—1062no—
129M16—2121no—
130C2×C8—41no—
131C2×C8—2121no—
132C4⋊C4—1062no—
133C4⋊C4—1062no—
134C4⋊C4—1061no—
135C22⋊C4—1061no—
136C22⋊C4—1062no—
137C22⋊C4—1062no—
138C42—532no—
139C23#1171992111yescharacteristic
140C23#1171992111yescharacteristic
141C23#1171992111yescharacteristic
142Q8—41no—
143Q8—2122no—
144Q8—2121no—
145Q8—2121no—
146D4—41no—
147D4—41no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—1062no—
156C2×C4—1062no—
157C2×C4—2122no—
158C2×C4#1171992111yescharacteristic
159C2×C4C2×C4×D5311yescharacteristic
160C2×C4C2×D21211yescharacteristic
161C2×C4C2×C22⋊D5311yescharacteristic
162C8—42no—
163C8—2122no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×#1171992111yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—2122no—
175C4—2122no—
176C4#1313313111yescharacteristic
177C4#1313313111yescharacteristic
178C2—21no—
179C2—41no—
180C2—41no—
181C2—41no—
182C2C2×#1313313111yescharacteristic
subgroupG/H
centerC2C2×#13133131
commutatorC2×C4×C53C22×C4
FrattiniC2×Q8C22×D53
FittingC53×#413C2
socleC2×C53C2×#99
hypercenter#413D53
hypocenterC53#2135
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #16810421C2semidirect
96. C53#2135semidirect